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Algebraic Geometry for non-mathematician

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I think I sound stupid but I have heard a lot about Algebraic Geometry as a subject and wish to study it without actually studying abstract algebra. I have never studied abstract algebra since I am a physics student. I have always studied mathematics as part of math physics course. Means I know basics of analysis and some of the concepts of algebra (groups, rings, fields). If I plan to study algebraic geometry for interest, what should be an ideally strategy. Should I master commutative algebra first ?

In my opinion commmutative algebra and algebraic geometry are so intertwined that the only way to study them is to study them toghether. I'd start with commutative algebra though (say at least until you understand well things like localizations and noetherianness)

The only "shortcut" I know of would be to play around with SURFER: imaginary.org/program/surfer. It shows you the roots of polynomials over 3 letters. Since varieties and algebraic curves are fundamental in AG this gives you a peek at the topic. Also an analytical statement about intersections (from eg lecture 1 of Vakil) can be envisaged with eg.

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@Henry.L: "for scientists and engineers" on my opinion means that scientists and engineers can also read it. A mathematical paper/paper which scientists and engineers can also read is something very good.

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You might be better off studying an analytic approach to algebraic geometry first. The classical reference for this approach to algebraic geometry is Griffith and Harris' Principles of Algebraic Geometry. Of course, to really understand algebraic geometry at a deeper level, you will need to acquire some algebraic tools.

If you are a physics student, you most certainly should study abstract algebra, particularly group theory. However, I would adopt an attitude of learning commutative algebra as needed at this point. Depending exactly what level of ability you wish to reach in algebraic geometry will determine how much commutative algebra you need to know. But for a first introduction, focus on the geometric aspects instead of the algebraic aspects, and Griffiths and Harris is a great place to start when this is your goal.

In addition, the OP might be better acquainted with complex geometry, so mastering related concepts from complex geometry (such as sheaves of functions, germs and say Dolbeault' cohomology) will be of value if and when he decides about doing the transition to the abstract algebraic setting.

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I would highly recommend first covering the book An Invitation to Algebraic Geometry by Smith et al. This requires very little abstract algebra as prerequisite (only understanding of very basics about rings) and gives a very well-written introduction to algebraic geometry. It has a great mix of theory and examples and gives a good idea of what algebraic geometry is and where it goes. As a plus, it has some good exercises asking the reader to work out some examples.

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Modern algebraic geometry is closely linked to commutative algebra. It is almost impossible to understand some of the essentials without knowing some algebra. Unfortunately, learning commutative algebra with no knowledge of geometry makes it a tediously dry subject. As a result, when learning commutative algebra, people wish they already knew algebraic geometry. And when learning algebraic geometry, people wish they already knew commutative algebra. There are a few books out there that try to deal with this. Here are two that I know of.

An introduction that begins essentially at the high school level (but gets to sheaves by the end) is the book by Tom Garrity et al, Algebraic Geometry: A Problem Solving Approach (American Mathematical Society, 2013). The book begins by looking at quadratic curves in the plane. It gets to sheaves and Cech cohomology by the end. Concepts such as ideals are introduced about halfway through. As the title says, this is essentially inquiry-based learning. So you will learn algebraic geometry by doing lots and lots of problems that build upon one another. [Disclaimer: I worked with Garrity et al on this book. I am surely biased in favor of it.]

A good book that starts at a higher level (say, an advanced undergraduate math major) while attempting to minimize the amount of commutative algebra needed is Introduction to Algebraic Geometry by Brendan Hassett (Cambridge University Press, 2007). Hassett's clever idea is to emphasize computation, in particular Gröbner bases, rather than building up the formal machinery of commutative algebra. You probably need to know some abstract algebra to get started, but not necessarily commutative algebra. Hassett gets up through Grassmannians, Hilbert polynomials, and such. There is a computational flair throughout the book.

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If you are comfortable doing a lot of exercises, and even better if you can work with a group, then I recommend Ravi Vakil's notes for his course, which are frequently updated and can be found here: http://math.stanford.edu/~vakil/216blog/ It is a hands-on treatment, and works through many facts you would need from commutative algebra and category theory.

From the introduction:

"Attempts to explain algebraic geometry often leave such background to the reader, refer to other sources the reader won’t read, or punt it to a telegraphic appendix. Instead, this book attempts to explain everything necessary, but as little possible, and tries to get across how you should think about (and work with) these fundamental ideas, and why they are more grounded than you might fear."

"The book is intended to be as self-contained as possible. I have tried to follow the motto: “if you use it, you must prove it”. I have noticed that most students are human beings: if you tell them that some algebraic fact is in some late chapter of a book in commutative algebra, they will not immediately go and read it. Surprisingly often, what we need can be developed quickly from scratch, and even if people do not read it, they can see what is involved."

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I always found the Harris book much more difficult than the two Shafarevich volumes. There are many examples in Harris's book, but there are also many claims that are left to the reader, and the prerequisites are quite substantial.

I would have put Harris's book last. My feeling is that one needs a lot of machinery to really understand a.g. Once one has mastered a lot of the abstract concepts, Harris's book provides numerous beautiful examples of actual varieties on which one can apply those tools.

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